(a) Find: ∫ x/(( x -1)( x^2+4)) dx

on solving we get, $\displaystyle A=\dfrac{1}{5}, B=\dfrac{-1}{5}, C=\dfrac{4}{5}$

so, $\displaystyle \int \dfrac{x}{(x-1)\left(x^{2}+4\right)} d x=\dfrac{1}{5} \int \dfrac{d x}{x-1}-\dfrac{1}{10} \int \dfrac{2 x d x}{x^{2}+4}+\dfrac{4}{5} \int \dfrac{d x}{x^{2}+4}$

$\displaystyle \int \dfrac{x}{(x-1)\left(x^{2}+4\right)} d x=\dfrac{1}{5} \log |x-1|-\dfrac{1}{10} \log \left(x^{2}+4\right)+\dfrac{2}{5} \tan^{-1}\left(\dfrac{x}{2}\right)+c$

$\displaystyle I$ (Let) $\displaystyle =\int_{0}^{1} \dfrac{x \tan^{-1} x}{\left(1+x^{2}\right)^{\frac{3}{2}}} d x=\int_{0}^{\frac{\pi}{4}} \dfrac{t \tan t}{\sec^{3} t} \sec^{2} t\, d t=\int_{0}^{\frac{\pi}{4}} t \sin t\, d t$

$\displaystyle I=(-t \cos t+\sin t)_{0}^{\frac{\pi}{4}}$

$\displaystyle I=\dfrac{-\pi}{4 \sqrt{2}}+\dfrac{1}{\sqrt{2}}$ or $\displaystyle \dfrac{-\pi+4}{4 \sqrt{2}}$